# Full text of "Treatise On Analysis Vol-Ii"

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```12   UNBOUNDED NORMAL OPERATORS       415

(ii) A closed vector subspace G of E x E is not the graph of a closed
operator if and only if, for some x e pr^G), there exist at least two distinct
points (x, y^ and (x, y2) belonging to G, or equivalently (since G is a vector
subspace) that (0, yl - y2) e G. But to say that dom(T*) is not dense in E
means that there exists z *£ 0 in E orthogonal to dom(T*) (6.3.1), or equiv-
alently that (z, 0) is orthogonal to r(T*), or again that (0, -z) belongs to
F(T). It follows that (0, — z) cannot be contained in the graph of any closed
operator which extends T. Conversely, if dom(T*) is dense in E, then T** is
defined, and T(T**) is the orthogonal supplement of /(r(T*)); but this ortho-
gonal supplement is also equal to

- r(T)

(6.3.1).    We say then that T** is the closure of T.

(15.12.5)    Given two not necessarily bounded operators U9 V on E, the
vector U • x + V • x is defined for all x E dom(£/) n dom(F), and we denote
by U + V the linear mapping xh-» U • x + V • x of dom(C7) n dom(F) into

E. In particular, if V is everywhere defined, then dom(£/~f- V) = dom(lT),
and the graph T(t/+ V) is the image of T(U) under the linear mapping
(x, y) h-» (jc, y + V-x)ofE x E into E x E. If V is continuous and U is closed,
it follows therefore that U 4- V is closed, since the mapping

(x,y)\->(x,y + V-x)

and its inverse (x, y) »-> (x, y — V • x) are continuous.

Again, the vector U • (V - x) is defined for the set of all x e E such that
xedom(F) and V* xe dom(£7). This set is a vector subspace which we
denote by dom(£/F), and UV denotes the linear mapping jc>~» U • (V • x) of
dom(£/F) into E. If T is a not necessarily bounded operator which is an
infective mapping of dom(T) into E, we denote by T"1 the inverse mapping of
T(dom(T)) = dom(T~1) into E. The graph HT'1) is the image of T(T)
under the mapping (x,y)\-*(y,x). Hence T""1 is closed if T is closed (and
injective).

(15.12.6)  (von Neumann) Let T be a closed operator on E such that dom(7) is
dense in E. Then dom(r*r) is dense in E; the operator T*T is closed; and the
operator I + T*T (defined on dom(r*7)) is a bijection of dom(r*T) onto E.
The operator B = (I+T*T)~1 is defined on E, continuous, self-adjoint and
injective, and its spectrum is contained in the interval [0, 1] of R. Furthermore,
the hermitian form (x, y))-+(B • x\y) is positive and nondegenerate, and C -
TB is a continuous operator defined on E, such that C(E) c: dom(r*). Finally,e restriction of G112
```